Question

Find the zeros of the function and state their multiplicities. y=x^4-8x^2+16

Answer 1

   
$y=0\\\\ x^4-8x^2+16 = 0\\\\ (x^2)^2 - 2\cdot 4 \cdot x^2 + 4^2 = 0\\\\ (x^2 - 4)^2 = 0\\\\ (x^2 - 4)(x^2 - 4)=0\\\\ (x-2)(x+2) (x-2)(x+2) = 0\\\\ (x-2)(x-2) (x+2)(x+2) = 0\\\\ \Longrightarrow~~~ x_1 = 2;~x_2 = 2;~ x_3= -2;~x_4 = -2\\\\ \text{Conclusion:}\\\\ x_1 = x_2 = 2~~~ \text{double root}\\\\ x_3 = x_4 = -2~~~ \text{double root}$